The Dirichlet boundary problems for second order
parabolic operators satisfying a Carleson condition
Sukjung Hwang
CMAC, Yonsei University
Collaboration with M. Dindos and M. Mitrea
The 1st Meeting of Young Researchers in PDEs
August 19, 2016
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Table of contents
1
Parabolic Equations with Lp −Dirichlet data
Carleson measure and Lipschitz domain
Non-tangential maximal functions
Pullback Transformation
Square and Area functions
2
L2 −Solvability for Parabolic equations
3
Solvaility for Elliptic system
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Elliptic equation
Definition of Solvability
For 1 < p ≤ ∞ and Ω (an admissible Lipschitz domain), consider the parabolic
Dirichlet boundary value problems

P

0 = i,j ∂xi aij (x)∂xj u(x) in Ω,
(1)
u = f ∈ Lp (∂Ω)
on ∂Ω,


p
N(u) ∈ L (∂Ω, dσ).
where A satisfies the uniform ellipticity conditions. We say (1) is solvable if there
exists a constant C such that
kN(u)kLp (∂Ω,dσ) ≤ C kf kLp (∂Ω,dσ)
where C depends on the operator, p, and the domain.
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Negative Results
Theorem
Consider
0=
X
∂xi aij (x)∂xj u(x) .
i,j
There exists a bounded measurable coefficient matrix A = [aij ] on a unit disk D
satisfying the uniform ellipticity conditions such that the Lp −Dirichlet problem is
not solvable for any p ∈ (1, ∞).
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Negative Results
Theorem
Consider
0=
X
∂xi aij (x)∂xj u(x) .
i,j
There exists a bounded measurable coefficient matrix A = [aij ] on a unit disk D
satisfying the uniform ellipticity conditions such that the Lp −Dirichlet problem is
not solvable for any p ∈ (1, ∞).
It is difficult because both the coefficients and the domain are given rough.
We impose minimal and natural smoothness conditions (so called Carleson
conditions) on the coefficients.
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Lp Dirichlet problems
Definition of Solvability
For 1 < p ≤ ∞ and Ω (an admissible Lipschitz domain), consider the parabolic
Dirichlet boundary value problems


ut = div (A∇u) + B · ∇u in Ω,
(2)
u = f ∈ Lp (∂Ω)
on ∂Ω,


p
N(u) ∈ L (∂Ω, dσ).
where A satisfies the uniform ellipticity conditions. We say (2) is solvable if there
exists a constant C such that
kN(u)kLp (∂Ω,dσ) ≤ C kf kLp (∂Ω,dσ)
where C depends on the operator, p, and the domain.
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Parabolic distance and ball
Parabolic distance:
δ(X , t) =
inf
|X − Y | + |t − s|1/2 .
(Y ,s)∈∂Ω
Parabolic ball:
Br (X , t) = {(Y , s) ∈ Rn × R : |X − Y | + |t − s|1/2 < r }.
Surface ball and Corresponding Interior:
∆r (X , t) = ∂Ω ∩ Br (X , t),
T (∆r )(X , t) = Ω ∩ Br (X , t).
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Carleson Measure
Definition
Then a measure µ : Ω → R+ is said to be Carleson if there exists a constant
C = C (r0 ) such that for all r ≤ r0 and all surface balls ∆r
µ (T (∆r )) ≤ C σ(∆r ).
The best constant C (r0 ) is called the Carleson norm.
If limr0 →0 C (r0 ) = 0, then we say that the measure µ satisfies the vanishing
Carleson condition.
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Assume that
2
2
δ(X , t)−1 oscBδ(X ,t)/2 (X ,t) aij + δ(X , t) supBδ(X ,t)/2 (X ,t) bi
is the density of Carleson measure with small Carleson norm.
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Assume that
2
2
δ(X , t)−1 oscBδ(X ,t)/2 (X ,t) aij + δ(X , t) supBδ(X ,t)/2 (X ,t) bi
is the density of Carleson measure with small Carleson norm.
Small Carleson conditions on the coefficients
|∇A|2 δ(X , t) + |At |2 δ 3 (X , t) dX dt,
|B|2 δ(X , t) dX dt
are Carleson measures with the norm .
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Lipschitz domain for Elliptic case
For X = (x0 , x) ∈ R ×Rn−1 , consider the region given by
Ω = {(x0 , x) : x0 > ψ(x)}
where the function ψ satisfies the followings
|ψ(x) − ψ(y )| ≤ L|x − y |.
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The time-varying Lipschitz domain
For X = (x0 , x) ∈ R ×Rn−1 , consider the region given by
Ω = {(x0 , x, t) : x0 > ψ(x, t)}
where the function ψ satisfies the followings
|ψ(x, t) − ψ(y , s)| ≤ L |x − y | + |t − s|1/2 .
Hunt’s conjecture: Lip(1, 1/2) is proper domain for parabolic equations
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The time-varying Lipschitz domain
For X = (x0 , x) ∈ R ×Rn−1 , consider the region given by
Ω = {(x0 , x, t) : x0 > ψ(x, t)}
where the function ψ satisfies the followings
|ψ(x, t) − ψ(y , s)| ≤ L |x − y | + |t − s|1/2 .
Hunt’s conjecture: Lip(1, 1/2) is proper domain for parabolic equations
Kaufmann & Wu (1988) : With only Lip(1, 1/2), parabolic measure and
surface measure are failed to be mutually absolutely continuous
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The admissible time-varying Lipschitz domain
For X = (x0 , x) ∈ R ×Rn−1 , consider the region given by
Ω = {(x0 , x, t) : x0 > ψ(x, t)}
where the function ψ satisfies the followings
|ψ(x, t) − ψ(y , s)| ≤ L |x − y | + |t − s|1/2 ,
t
kD1/2
ψkBMO ≤ L.
Lewis & Murray (1995) : Establish mutual absolute continuity of caloric
measure and a certain parabolic analogue of surface measure
An admissible parabolic is defined locally satisfying Lip(1, 1/2) and
‘Lewis-Murray’ conditions using at most N L−cylinders.
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Non-tangential maximal functions
For (x0 , x, t) ∈ ∂Ω, define a parabolic cone
Γa (x0 , x, t) = {(y0 , y , s) ∈ Ω : |x − y | + |s − t|1/2 ≤ a(y0 − x0 ), y0 > x0 }.
Non-tangential maximal functions:
Na (u)(x0 , x, t) =
sup
|u(y0 , y , s)| .
(y0 ,y ,s)∈Γa (x0 ,x,t)
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Property of Carleson Measure
Non-tangential maximal function and Carleson measure
We have integral inequalities that
ˆ
δ|∇A|2 + δ 3 |At |2 + δ|B|2 u 2 dX dt
T (∆r )
ˆ
≤ kµkC
N 2 (u) dx dt
∆r
where kµkC is a Carleson norm.
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Dahlberg (1977): Elliptic problem with L2 Dirichlet boundary data
Kenig, Koch, Pipher, Toro (2000): Elliptic operator satisfying Carleson
measure condition, Comparison of the square and non-tangential maximal
functions
Dindos, Petermichl, Pipher (2007): Lp Dirichlet Elliptic problems under
small Carleson conditions, Quantitative approach
Symmetry of the coefficients is NOT assumed.
Does NOT rely on layer potentials rather direct approach.
Compare the non-tangential maximal function and the square function
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Pullback Transformation
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Pullback Transformation
x0 independent transformation
It is trivial to use
ρ(x0 , x, t) = (x0 + ψ(x, t), x, t).
But vt provides ψt (x, t)ux0 (x0 , x, t) where ψt may not defined because of
the lack of regularity.
For Elliptic equations, corresponding coefficient A is x0 independent. (Kenig,
Koch, Pipher, & Toro 2000, 2 dimension for Dirichlet problem, Hofmann,
Kenig, Mayboroda, & Pipher Higher dimensional Dirichlet problem, Rellich
inequality)
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Dalhberg-Kenig-Necas-Stein transformation using mollification
We define
ρ(x0 , x, t) = (x0 + Pγx0 ψ(x, t), x, t) .
For a nonnegative function P(x, t) ∈ C0∞ for (x, t) ∈ Rn−1 × R, set
Pλ (x, t) ≡ λ−(n+1) P(λ−1 x, λ−2 t),
ˆ
Pλ ψ(x, t) ≡
Pλ (x − y , t − s)ψ(y , s) dy ds.
Rn−1 ×R
Elliptic Cases: basically allows Layer potentials (Mitrea & Taylor 2003,
Hofamman & Lewis 2001 Double layer potentials for parabolic equations)
We avoid layer potential methods.
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Definitions
Square functions:
ˆ
!1/2
|∇u|2 δ −n (y0 , y , s) dy0 dy ds
Sa (u)(x0 , x, t) =
,
Γa (x0 ,x,t)
ˆ
kSa (u)k2L2 (∂Ω) =
|∇u|2 δ(y0 , y , s) dy0 dy ds
Ω
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Definitions
Square functions:
ˆ
!1/2
|∇u|2 δ −n (y0 , y , s) dy0 dy ds
Sa (u)(x0 , x, t) =
,
Γa (x0 ,x,t)
ˆ
kSa (u)k2L2 (∂Ω) =
|∇u|2 δ(y0 , y , s) dy0 dy ds
Ω
Area functions:
ˆ
!1/2
2 −n+2
|ut | δ
Aa (u)(x0 , x, t) =
(y0 , y , s) dy0 dy ds
,
Γa (x0 ,x,t)
ˆ
kAa (u)k2L2 (∂Ω) =
|ut |2 δ 3 (y0 , y , s) dy0 dy ds.
Ω
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Strategy
L∞ : the Maximum Principle
Lp for 2 < p < ∞ : Interpolation if we have L2 −solvability
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Strategy
L∞ : the Maximum Principle
Lp for 2 < p < ∞ : Interpolation if we have L2 −solvability
L2 : Quantitative method comparing kS(u)kL2 (∂Ω) and kN(u)kL2 (∂Ω)
Lp for 1 < p < 2: Quantitative method with p−adapted square and area
functions.
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L2 Dirichlet problems
Definition of Solvability
For Ω (an admissible Lipschitz domain), consider the parabolic Dirichlet boundary
value problems

ut = div (A∇u) + B · ∇u in Ω,

(3)
u = f ∈ L2 (∂Ω)
on ∂Ω,


2
N(u) ∈ L (∂Ω, dσ).
where A satisfies the uniform ellipticity conditions. We say (3) is solvable if there
exists a constant C such that
kN(u)kL2 (∂Ω,dσ) ≤ C kf kL2 (∂Ω,dσ)
where C depends on the operator and the domain.
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L2 Dirichlet problem
Uniform Ellipticity and Boundedness for A = [aij ]
X
λ|ξ|2 ≤
aij ξi ξj ≤ Λ|ξ|2 .
Small Carleson conditions on the coefficients
|∇A|2 δ(X , t) + |At |2 δ 3 (X , t) dX dt,
|B|2 δ(X , t) dX dt
are Carleson measures with the norm .
Boundedness from the Carleson conditions
|∇A|δ(X , t) + |At |δ 2 (x, t) ≤ 1/2 ,
|B|δ(X , t) ≤ 1/2 .
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Rough Sketch
It is the results of three estimates.
Comparing A(u) and S(u) using Caccioppoli’s inequality
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Rough Sketch
It is the results of three estimates.
Comparing A(u) and S(u) using Caccioppoli’s inequality
First,
ˆ
ˆ
S 2 (u) dx dt ≤ C
∂Ω
ˆ
f 2 dx dt + ∂Ω
N 2 (u) dx dt
∂Ω
where comes from the smallness of the Carleson measure of the
coefficients.
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Rough Sketch
It is the results of three estimates.
Comparing A(u) and S(u) using Caccioppoli’s inequality
First,
ˆ
ˆ
ˆ
S 2 (u) dx dt ≤ C
∂Ω
f 2 dx dt + ∂Ω
N 2 (u) dx dt
∂Ω
where comes from the smallness of the Carleson measure of the
coefficients.
The second inequality is given by
ˆ
ˆ
N 2 (u) dx dt ≤ C
∂Ω
Hwang
ˆ
S 2 (u) dx dt +
∂Ω
Carleson condition
f 2 dx dt .
∂Ω
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We replace kS(u)kL2 (∂Ω) by the first inequality. Then
ˆ
ˆ
ˆ
2
2
2
N (u) dX dt ≤ C
S (u) dX dt +
f dX dt
∂Ω
ˆ ∂Ω
ˆ ∂Ω
≤ C0
f 2 dX dt + N 2 (u) dX dt.
∂Ω
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∂Ω
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We replace kS(u)kL2 (∂Ω) by the first inequality. Then
ˆ
ˆ
ˆ
2
2
2
N (u) dX dt ≤ C
S (u) dX dt +
f dX dt
∂Ω
ˆ ∂Ω
ˆ ∂Ω
≤ C0
f 2 dX dt + N 2 (u) dX dt.
∂Ω
∂Ω
Because of , we can hide kN(u)kL2 (∂Ω) . Therefore, it follows
kN(u)kL2 (∂Ω) ≤ C kf kL2 (∂Ω) .
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Theorem
For an admissible parabolic domain Ω, let A = [aij ] satisfy the uniform ellipticity
and boundedness with constants λ and Λ. In addition, assume that
"
#
2
dµ = δ(X , t)−1 sup
1≤i,j≤n
osc
Bδ(X ,t)/2 (X ,t)
aij
+ δ(X , t)
sup
|B|
2
dX dt
Bδ(X ,t)/2 (X ,t)
is the density of a Carleson measure on Ω with Carleson norm kµkC .
Continue..
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Theorem
Then there exists ε > 0 such that if for some r0 > 0 max{L, kµkC ,r0 } < ε then
the Lp boundary value problem


ut = div(A∇u) + B · ∇u in Ω,
(4)
u = f ∈ Lp
on ∂Ω,


p
N(u) ∈ L (∂Ω),
is solvable for all 1 ≤ p < ∞. Moreover, the estimate
kN(u)kLp (∂Ω,dσ) ≤ Cp kf kLp (∂Ω,dσ) ,
(5)
holds with Cp = Cp (L, kµkC ,r0 , N, λ, Λ).
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Solvability for Elliptic system
Joint work with M. Dindos and M. Mitrea (U of Missouri)
Provide quantitative method for Elliptic system with L2 −Dirichlet boundary
condition.
The lack of regularity such as no Maximum principle. Not always possible to
obatin Lp solvability for p > 2.
With Regularity problem and Reverse Hölder inequality, in fact, we have
results on Lp −Dirichlet problem for 2 ≤ p < p ∗ where p ∗ depending on n
and p.
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Definition
Let Ω be the Lipschitz domain {(x0 , x 0 ) : x0 > φ(x 0 )}. Let 1 < p < ∞. Consider
the following Dirichlet problem for a vector valued function u : Ω → Rn

h i
αβ
αβ


0 = ∂i Aij (x)∂j uβ + Bi (x)∂i uβ α in Ω, α = 1, 2, . . . , N
(6)
u(x) = f (x)
on ∂Ω,


Ñ(u) ∈ Lp (∂Ω).
Here we use the usual Einstein convention and sum over repeating indices (i and
β). We say the Dirichlet problem (6) is solvable for a given p ∈ (1, ∞), if there
exists C = C (λ, Λ, n, p, Ω) > 0 such that the unique energy solution
(u ∈ W 1,2 (Ω) obtained via Lax-Milgram lemma) with boundary data
2,2
f ∈ Lp (∂Ω) ∩ B1/2
(∂Ω) satisfies the estimate
kÑukLp (∂Ω) ≤ C kf kLp (∂Ω) .
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(7)
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Definition
For Ω ⊂ Rn as above the nontangential maximal function of u at Q ∈ ∂Ω relative
to the cone Γa (Q) and its truncated version at height h are defined by
Na (u)(Q) = sup |u(x)| and Nah (u)(Q) =
x∈Γa (Q)
sup |u(x)|.
x∈Γha (Q)
Moreover, we shall also consider a weaker version Ñ of the nontangential maximal
function that is defined using L2 averages over balls in the domain Ω. We set
Ña (u)(Q) = sup w (x)
and Ñah (u)(Q) =
x∈Γa (Q)
where
w (x) :=
Hwang
sup w (x)
x∈Γha (Q)
ˆ
1
!1/2
2
|Bδ(x)/2 (x)|
u (z) dz
.
(8)
Bδ(x)/2 (x)
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Theorem (assumptions)
Let Ω be the Lipschitz domain {(x0 , x 0 ) : x0 > φ(x 0 )} with Lipschitz constant
L = k∇φkL∞ . Assume that the coefficients A of the system (6) are strongly
elliptic with parameters λ, Λ . In addition assume that at least one of the
following holds.
= IN×N on Ω.
(i) Ω = Rn+ and the matrix A00 = Aαβ
00
α,β
(ii) For all i, j, α, β we have
Hwang
Aαβ
ij
=
Aβα
ij
on Ω.
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Theorem (Results)
Then there exists K = K (λ, Λ, n) > 0 such that if

!
!2 
2
dµ(x) = 
|∇A(x)|
sup
Bδ(x)/2 (x)
+
sup
|B(x)|
 δ(x) dx
(9)
Bδ(x)/2 (x)
is a Carleson measure with norm kµkC less than K and the Lipschitz constant
L < K the the L2 Dirichlet problem for the system (6) is solvable and the estimate
kÑukL2 (∂Ω) ≤ C kf kL2 (∂Ω) .
2,2
holds for all energy solutions u with f ∈ L2 (∂Ω) ∩ B1/2
(∂Ω) where
C (λ, Λ, n, Ω) > 0.
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Thank you for your attention!!
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